Problem: Determine the smallest positive integer $n$ such that $5^n\equiv n^5\pmod 3$.
Answer: First notice that $5^n\equiv 2^n\pmod 3$, which should make our computations easier. For $n=1,2,3,4$, we get $2^n\equiv 2,1,2,1\pmod 3$ respectively and $n^5\equiv 1,2,0,1\pmod 3$ respectively. Since we have a congruent pair at $n=\boxed{4}$, we don't need to look any further.